While learning topology one learns about compact set. The standard definition is:
- A set $X$ is said to be compact if open cover has a finite subcover.
Since $[0,1]$ is compact, if we take a open cover for this we should be able to get a finite subcover. I know, that $(0,1)$ is not compact, so there must exists some open cover for $(0,1)$ which doesn't admit any finite subcover. But how does one prove this fact?
Best Answer
Isn't this clear from $(0,1) = (0,1/2) \cup (0,2/3) \cup (0,3/4) \cup \cdots$?